25 problems

Acyclicity conjecture. All $d$-representation finite algebras have acyclic quivers if and only if all twisted fractionally Calabi–Yau algebras have acyclic quivers.

Higher Koszul cohomology Calabi–Yau conjecture. Then $A$ is $n$-Calabi–Yau. This conjecture proposes that concentration of the higher Koszul cohomology in a single degree, with one…

Let $R$ be the base algebra, let $A$ be a friendly Calabi–Yau $R$-algebra of dimension $3$, let $F$ be a smooth $R$-algebra, and let…

Let $(D,\omega,\xi)$ be the symplectic data used in the universal construction, and let $A$ be a friendly algebra satisfying the stated Calabi–Yau condition. Theorem main(ii) repre…

Dubois-Violette's conjecture. Any $3$-regular multilinear form provides an algebra in $\mathcal{C}_2$.

Van den Bergh's conjecture. Any three-dimensional Calabi–Yau algebra derives from a potential.

Acyclic Ext-quiver conjectures. 1. If $A$ is twisted fractionally Calabi–Yau, then $A$ has an acyclic Ext-quiver. 2. If $A$ is $d$-representation-finite, then $A$ has an acyclic Ex…

Let $\mathbb{L}$ be a compact, relatively spin, graded, oriented immersed Lagrangian of dimension $n$ in a symplectic manifold $M$, and let $\mathcal{A}_{\mathbb{L}}$ be its Maurer…

Twisted Calabi–Yau conjecture. $H(\alpha,\beta,\gamma)$ is twisted Calabi–Yau in general.

Let $A$ be an indecomposable twisted Calabi–Yau algebra. Primality conjecture. Every indecomposable twisted Calabi–Yau algebra is prime. The source presents this as a conjecture fo…

A $3$-Calabi--Yau algebra is an algebra with the corresponding Calabi--Yau property, and a quiver with potential consists of a quiver together with a potential whose associated Gin…

Let $X$ be a compact, orientable $3$-dimensional manifold with contractible universal cover, and let $k[pi_1(X)]$ be its fundamental group algebra. A superpotential algebra is an a…

Multiplicative preprojective algebra conjecture. For every non-Dynkin quiver, the corresponding properties—including 2-Calabi–Yau-ness, primeness, flatness in $q$, and the asserted…

Let $Q$ be a connected non-Dynkin quiver and let $q\in (\mathbb{C}^\times)^{Q_0}$. Write $\Lambda^q(Q)$ for the multiplicative preprojective algebra of $Q$ with parameter $q$. 2-Ca…

Let $A$ be a locally finite graded twisted Calabi–Yau algebra of dimension $d$, and suppose that $A$ has finite Gelfand–Kirillov dimension. GK-dimension conjecture. The Gelfand–Kir…

Ginzburg's conjecture. All 3-dimensional Calabi–Yau algebras should arise as Jacobi algebras. This conjecture concerns a proposed description of all such algebras by quivers with p…

Rouquier's conjecture. The isomorphism

CY property conjecture. These universal quantum groups should possess the same homological properties as $A$, in particular the Calabi–Yau property.

A $d$-Calabi–Yau algebra $A$ is an algebra satisfying suitable finiteness conditions together with a duality condition involving Hochschild cohomology. A potential is a non-commuta…

Let $M$ be an acyclic compact orientable manifold, let $\pi(M)$ denote its fundamental group, and let $k[\pi(M)]$ be the corresponding group algebra. Circle-bundle superpotential c…

Let $M$ be a $d$-dimensional compact orientable acyclic manifold, and let $k[\pi_1(M)]$ be its fundamental group algebra. Write $\operatorname{HH}_d(k[\pi_1(M)])$ for its degree-…

Let $M$ be an acyclic oriented compact manifold of dimension $d$, and let $k[\pi_1(M)]$ denote its fundamental group algebra. Superpotential conjecture. If $M$ is an acyclic orient…

Artin–Schelter Gorenstein–twisted Calabi–Yau conjecture. For all connected graded noetherian rings, the Artin–Schelter Gorenstein property is equivalent to the twisted Calabi–Yau p…

Calabi–Yau Hochschild pairings conjecture. For any Calabi–Yau DG algebra $A$, the two pairings coincide: for all $a,b\in\mathsf{HH}_\bullet(A)$,

Let $\Phi$ be a homogeneous Calabi–Yau potential of degree $d=a+b+c$, let $\Psi\in{\mathfrak A}(\Phi)$ be a homogeneous central element of degree $d$, and let $P$ be a point module…